T(n,k)=Half the number of 0..k arrays of length n+2 with second differences nonzero.

Terms

a(0) =3a(1) =11a(2) =5a(3) =28a(4) =27a(5) =8a(6) =56a(7) =99a(8) =66a(9) =13a(10) =99a(11) =252a(12) =350a(13) =162a(14) =21a(15) =159a(16) =546a(17) =1134a(18) =1238a(19) =397a(20) =34a(21) =240a(22) =1034a(23) =3010a(24) =5104a(25) =4379a(26) =973a(27) =55a(28) =344a(29) =1803